In particular, any \(\mathcal {H}^1\) -rectifiable set \(K\)  admits a approximate tangent line at \(\mathcal {H}^1\) -a.e. \(x\in K,\)  see e.g., [1]}. When \(\theta _*(K,x)=\theta ^*(K,x)=1,\)  we write for simplicity \(\theta (K,x)=1.\) 
A Borel set \(K\subset \mathbb {R}^2\)  with \(\mathcal {H}^1(K)<+\infty \)  is said purely unrectifiable if \(\mathcal {H}^1(K\cap \Gamma )\)  for every 1-dimensional Lipschitz graph \(\Gamma \subset \mathbb {R}^2\)  (see e.g., [2]}).
